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Math 4361/52 T4st 1 February 10, 2016
Part I: Do on this page. Answers are sufficient.
1. (2 pts each) Mark the blank for each of the following T, if the statement is true, and
F, if the statement is false.
(a)
F
(b)
If
*
is a binary operation on a set S, then a * a = a for all a in S
A binary operation
in Sso that a * b = b*a.
(c)
T
(d)
fr
(e)
T
()r
(g) )
*
.
on a set S is commutative provided there are a, b
The associative law holds in every group.
There may be a group in which the cancellation law fails.
Any two groups with three elements are isomorphic.
Every group with at most three elements is abelian.
The empty set can be considered a group.
(h)T' Z4 is a cyclic group.
(i)
(
Every set of numbers that is a group under addition is also a group
under multiplication.
Every subset of a group is a subgroup under the induced operation.
(j)
As additive groups, Z and 2Z are isomorphic.
k)--7— As
(l
(m)
TJ
(n)
(o)
There is at least one abelian group of every order > 0.
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All generators of Z20 are prime numbers.
Every cyclic group of order > 2 has at least two generators.
An equation of the form a * x * b = c always has a unique solution x in
a group.
Part II
Do these here, if you have room, or use the supplied paper. Show whatever work is required
to do each problem.
1. (5 pts) Find all solutions to x+7x=3inZ7.
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2, (5 pts) What is the gcd of 48 and 88?
3. (5 pts) How many generators does a cyclic group with order 12 have?
4. (5 pts) What is the order of the cyclic subgroup of Z30 generated by 25?
5. (5 pts) How many proper, nontriviajjsubgroups does Z6 have?
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6. (5 pts) Give an example of a nonabelian group. Specify the operation.
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Math 4361/52 Tst 1 February 10, 2016
Part I: Do on this page. Answers are sufficient.
1. (2 pts each) Mark the blank for each of the following T, if the statement is true, and
F, if the statement is false.
(a)F
If * is a binary operation on a set S, then a * a
=
a for all a in S.
(b) j
A binary operation * on a set S is commutative provided there are a, b
in Sso that a*b=b*a.
(c) T
(d)
F
The associative law holds in every group.
There may be a group in which the cancellation law fails.
(e)
Any two groups with three elements are isomorphic.
()r
(g) )
Every group with at most three elements is abelian.
(h)T
(i)
The empty set can be considered a group.
Z4 is a cyclic group.
Every set of numbers that is a group under addition is also a group
under multiplication.
(j) T
Every subset of a group is a subgroup under the induced operation.
(k)T
As additive groups, Z and 2Z are isomorphic.
(l)T
There is at least one abelian group of every order > 0.
(m)
i-S
(n)
T
(o)
I
All generators of Z20 are prime numbers.
2
(
:
Every cyclic group of order > 2 has at least two generators.
An equation of the form a * x * b
=
c always has a unique solution x in
Part II
Do these here, if you have room, or use the supplied paper. Show whatever work is required
to do each problem.
1. (5 pts) Find all solutions to x +7 x
=
I)
3 in Z7.
2.(5 pts) What is the gcd of 48 and 88?
3.(5 pts) How many generators does a cyclic group with order 12 have?
4. (5 pts) What is the order of the cyclic .subgroup of Z30 generated by 25?
5. (5 pts) How many proper, nontrivialjsubgroups does
(2t
4 have?
6. (5 pts) Give an example of a nonabelian group. Specify the operation.
(23o)
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Part III
Write fully supported answers for the following.
1. (10 pts) Let H be the subset of M2(R) consisting of all matrices of the form
a —b
b
a
(a) List three different elements in H.
(b) Is H closed under addition?
-
2. (10 pts) Let r, s be positive integers. Show that H
subgroup of Z.
3. (10 pts) Show that if a2 * b2
=
=
{rn + sin I ri, rn
E
Z} is a
(a * b)2 for all a, b in a group C, then C is abelian.
4. (10 pts) Show that if H and K are subgroups of a group C, then HflK is a subgroup
of G.
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